Abelian variety isogeny class 4.3.g_x_ci_er over $\F_{3}$

• Label: 4.3.g_x_ci_er
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 + 3 x + 7 x^{2} + 9 x^{3} + 9 x^{4} )^{2}$
	$1 + 6 x + 23 x^{2} + 60 x^{3} + 121 x^{4} + 180 x^{5} + 207 x^{6} + 162 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.535169663346$, $\pm0.535169663346$, $\pm0.772732979144$, $\pm0.772732979144$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$29$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$841$	$21025$	$303601$	$42575625$	$3152148736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$20$	$10$	$84$	$220$	$860$	$2110$	$6084$	$20710$	$58850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x^2+y^2+z t=y^3+x z^2-x t^2-y t^2=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 2.3.d_h^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1525.1$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
4.3.ag_x_aci_er	$2$	(not in LMFDB)
4.3.a_f_a_n	$2$	(not in LMFDB)
4.3.ad_c_ad_n	$3$	(not in LMFDB)